GAM evaluation - SF Bay Delta station C3
Marcus Beck
Observed data
Observed log-chlorophyll at representative station in SF Bay Delta region.
library(tidyverse)
library(lubridate)
library(mgcv)
library(plotly)
library(WRTDStidal)
library(gridExtra)
source('R/funcs.R')
# flow data, left moving window average of 30 days
data(sf_fldat)
fl_dat <- sf_fldat %>%
rename(date = Date) %>%
filter(station %in% 'sac') %>%
mutate(
qsm = stats::filter(q, rep(1, 30)/30, sides = 1, method = 'convolution')
)
# format the data to model
data(sf_dat)
sf_mod <- sf_dat %>%
filter(Site_Code %in% 'C3') %>%
rename(date = Date) %>%
mutate(
doy = yday(date),
dec_time = decimal_date(date),
yr = year(date),
mo = month(date, label = T)
) %>%
left_join(fl_dat, by = 'date') %>%
mutate( # all variables ln-transformed
flo = log(qsm),
chl = log(chl),
tss = log(tss),
nh = log(nh)
) %>%
select(-q, -qsm, -station, -Latitude, -Longitude, -Location)
# plot, all
p <- ggplot(sf_mod, aes(x = date, y = chl)) +
geom_line() +
theme_bw()
ggplotly(p)# boxplot, by years
p <- ggplot(sf_mod, aes(x = yr, y = chl)) +
geom_boxplot() +
theme_bw()
ggplotly(p)# boxplot, by month
p <- ggplot(sf_mod, aes(x = mo, y = chl)) +
geom_boxplot() +
theme_bw()
ggplotly(p)GAMs with annual, seasonal trends
Some simple GAMs to explore annual, seasonal trends.
# smooths to evaluate
smths <- c(
"s(dec_time, bs = 'tp')",
"s(doy, bs = 'cc')",
"te(dec_time, doy, bs = c('tp', 'cc'))"
)
# get all combinations of smoothers to model, one to many
frms <- list()
for(i in seq_along(smths)){
frm <- combn(smths, i) %>%
apply(2, function(x){
paste(x, collapse = ' + ') %>%
paste('chl ~ ', .) %>%
formula
})
frms <- c(frms, frm)
}
# create models from smooth formula combinations
mods <- map(frms, function(frm){
gam(frm,
knots = list(doy = c(1, 366)),
data = sf_mod,
na.action = na.exclude
)
})
names(mods) <- paste0('mod', seq_along(mods))Summary stats of annual, seasonal models:
# smoother stats of GAMs
map(mods, ~ summary(.x)$s.table %>% data.frame %>% rownames_to_column('smoother')) %>%
enframe %>%
unnest %>%
kable(digits = 2)| name | smoother | edf | Ref.df | F | p.value |
|---|---|---|---|---|---|
| mod1 | s(dec_time) | 6.87 | 7.96 | 29.96 | 0 |
| mod2 | s(doy) | 3.19 | 8.00 | 12.60 | 0 |
| mod3 | te(dec_time,doy) | 13.36 | 15.97 | 27.26 | 0 |
| mod4 | s(dec_time) | 7.27 | 8.28 | 35.57 | 0 |
| mod4 | s(doy) | 3.59 | 8.00 | 18.28 | 0 |
| mod5 | s(dec_time) | 7.35 | 8.33 | 18.75 | 0 |
| mod5 | te(dec_time,doy) | 8.79 | 15.00 | 11.09 | 0 |
| mod6 | s(doy) | 2.93 | 8.00 | 2.36 | 0 |
| mod6 | te(dec_time,doy) | 10.64 | 12.98 | 22.28 | 0 |
| mod7 | s(dec_time) | 7.33 | 8.31 | 9.28 | 0 |
| mod7 | s(doy) | 3.39 | 8.00 | 5.28 | 0 |
| mod7 | te(dec_time,doy) | 6.64 | 15.00 | 1.42 | 0 |
# summary stats of GAMs
map(mods, ~ data.frame(
AIC = AIC(.x),
R2 = summary(.x)$r.sq)) %>%
enframe %>%
unnest %>%
kable(digits = 2)| name | AIC | R2 |
|---|---|---|
| mod1 | 1117.92 | 0.30 |
| mod2 | 1215.82 | 0.16 |
| mod3 | 995.10 | 0.45 |
| mod4 | 986.88 | 0.45 |
| mod5 | 972.21 | 0.47 |
| mod6 | 992.15 | 0.45 |
| mod7 | 971.92 | 0.48 |
# prediction data
pred_dat <- data.frame(
dec_time = seq(min(sf_mod$dec_time), max(sf_mod$dec_time), length = 1000)
) %>%
mutate(
date = date_decimal(dec_time),
date = as.Date(date),
mo = month(date, label = TRUE),
doy = yday(date),
yr = year(date)
) %>%
left_join(., fl_dat[, c('date', 'qsm')]) %>%
mutate(flo = log(qsm)) %>%
select(-qsm)
# predictions
sf_res <- map(mods, function(x){
pred_dat %>%
mutate(
pred = predict(x, newdata = pred_dat)
)
}) %>%
enframe('mods') %>%
unnest
# plot
p <- ggplot(sf_res, aes(x = date)) +
geom_point(data = sf_mod, aes(y = chl), size = 0.5) +
geom_line(aes(y = pred, colour = mods)) +
theme_bw() +
theme(
legend.position = 'top',
legend.title = element_blank()
)
ggplotly(p)# plot
p <- ggplot(sf_res, aes(x = doy, group = factor(yr), colour = yr)) +
geom_line(aes(y = pred)) +
theme_bw() +
theme(
legend.position = 'top',
legend.title = element_blank()
) +
facet_wrap(~ mods, ncol = 2)
ggplotly(p)GAMs with annual, seasonal, and flow trends
Adding flow data to the model:
# smooths to evaluate
smths <- c(
"s(dec_time, bs = 'tp')",
"s(doy, bs = 'cc')",
"s(flo, bs = 'tp')",
"te(flo, doy, bs = c('tp', 'cc'))",
"te(flo, dec_time, bs = c('tp', 'tp'))",
"te(dec_time, doy, bs = c('tp', 'cc'))",
"te(dec_time, doy, flo, bs = c('tp', 'cc', 'tp'))"
)
# get all combinations of smoothers to model, one to many
frms <- list()
for(i in seq_along(smths)){
frm <- combn(smths, i) %>%
apply(2, function(x){
paste(x, collapse = ' + ') %>%
paste('chl ~ ', .) %>%
formula
})
frms <- c(frms, frm)
}
# create models from smooth formula combinations
mods2 <- map(frms, function(frm){
gam(frm,
knots = list(doy = c(1, 366)),
data = sf_mod,
na.action = na.exclude
)
})
names(mods2) <- paste0('mod', seq_along(mods2))
save(mods2, file = 'data/sf_mods2.RData', compress = 'xz')Summary stats of best year/season model, year/season/flow model
data(sf_mods2)
# best model with only season, year
best1 <- map(mods, ~ summary(.x)$r.sq) %>%
unlist %>%
which.max %>%
mods[[.]]
# best model with season, year, flow
best2 <- map(mods2, ~ summary(.x)$r.sq) %>%
unlist %>%
which.max %>%
mods2[[.]]
best <- list(best1 = best1, best2 = best2)
# smoother stats of GAMs
map(best, ~ summary(.x)$s.table %>% data.frame %>% rownames_to_column('smoother')) %>%
enframe %>%
unnest %>%
kable(digits = 2)| name | smoother | edf | Ref.df | F | p.value |
|---|---|---|---|---|---|
| best1 | s(dec_time) | 7.33 | 8.31 | 9.28 | 0.00 |
| best1 | s(doy) | 3.39 | 8.00 | 5.28 | 0.00 |
| best1 | te(dec_time,doy) | 6.64 | 15.00 | 1.42 | 0.00 |
| best2 | s(dec_time) | 6.45 | 7.55 | 4.65 | 0.00 |
| best2 | s(flo) | 3.20 | 4.05 | 1.93 | 0.09 |
| best2 | te(flo,dec_time) | 11.20 | 16.00 | 1.85 | 0.00 |
| best2 | te(dec_time,doy,flo) | 37.05 | 75.00 | 3.69 | 0.00 |
# summary stats of GAMs
map(best, ~ data.frame(
AIC = AIC(.x),
R2 = summary(.x)$r.sq)) %>%
enframe %>%
unnest %>%
kable(digits = 2)| name | AIC | R2 |
|---|---|---|
| best1 | 971.92 | 0.48 |
| best2 | 880.84 | 0.59 |
# predictions
sf_res2 <- map(best, function(x){
pred_dat %>%
mutate(
pred = predict(x, newdata = pred_dat)
)
}) %>%
enframe('mods') %>%
unnest
# plot
p <- ggplot(sf_res2, aes(x = date)) +
geom_point(data = sf_mod, aes(y = chl), size = 0.5) +
geom_line(aes(y = pred, colour = mods)) +
theme_bw() +
theme(
legend.position = 'top',
legend.title = element_blank()
)
ggplotly(p)ptheme <- theme(
axis.title.x = element_blank(),
axis.title.y = element_blank()
)
cols <- 'Spectral'
pb1 <- dynagam(best1, pred_dat, ncol = 1, col_vec = cols) +
ptheme +
theme(legend.position = 'none') +
ggtitle('Best 1')
pb2 <- dynagam(best2, pred_dat, ncol = 1, col_vec = cols) +
ptheme +
ggtitle('Best2')
pleg <- g_legend(pb2)
pb2 <- pb2 +
theme(legend.position = 'none')
grid.arrange(
pleg,
arrangeGrob(pb1, pb2, ncol = 2, bottom = 'lnQ', left = 'lnchl'),
ncol = 1,
heights = c(0.1, 1)
)
Adding nitrogen and turbidity covariates
# formula for best annual, seasonal, flow model
strt <- best2$formula %>%
as.character
smths <- c(
"s(nh, bs = 'tp')",
"s(tss, bs = 'tp')",
"te(nh, tss, bs = c('tp', 'tp'))"
)
# get all combinations of smoothers to model, one to many
frmstab <- list()
frms <- list()
for(i in seq_along(smths)){
# for the summary table
frmtab <- combn(smths, i) %>%
apply(2, function(x){
paste(x, collapse = ' + ')
})
# for the model
frm <- sapply(frmtab, function(x){
paste('chl ~', strt[3], '+', x) %>%
formula
})
frmstab <- c(frmstab, frmtab)
frms <- c(frms, frm)
}
# create models from smooth formula combinations
mods3 <- map(frms, function(frm){
gam(frm,
knots = list(doy = c(1, 366)),
data = sf_mod,
na.action = na.exclude
)
})
names(mods3) <- paste0('mod', seq_along(mods3))Summary of all nutrient, tss models:
# smoother stats of GAMs
map(mods3, ~ summary(.x)$s.table %>% data.frame %>% rownames_to_column('smoother')) %>%
enframe %>%
unnest %>%
kable(digits = 2)| name | smoother | edf | Ref.df | F | p.value |
|---|---|---|---|---|---|
| mod1 | s(dec_time) | 6.73 | 7.79 | 7.44 | 0.00 |
| mod1 | s(flo) | 3.10 | 3.96 | 3.09 | 0.01 |
| mod1 | te(flo,dec_time) | 11.10 | 16.00 | 1.74 | 0.00 |
| mod1 | te(dec_time,doy,flo) | 36.51 | 75.00 | 3.74 | 0.00 |
| mod1 | s(nh) | 1.00 | 1.00 | 10.76 | 0.00 |
| mod2 | s(dec_time) | 6.76 | 7.81 | 3.57 | 0.00 |
| mod2 | s(flo) | 4.56 | 5.61 | 0.52 | 0.81 |
| mod2 | te(flo,dec_time) | 6.31 | 16.00 | 0.87 | 0.00 |
| mod2 | te(dec_time,doy,flo) | 29.49 | 75.00 | 3.27 | 0.00 |
| mod2 | s(tss) | 4.46 | 5.53 | 6.33 | 0.00 |
| mod3 | s(dec_time) | 7.16 | 8.16 | 3.02 | 0.00 |
| mod3 | s(flo) | 1.00 | 1.00 | 1.01 | 0.31 |
| mod3 | te(flo,dec_time) | 6.71 | 16.00 | 0.92 | 0.00 |
| mod3 | te(dec_time,doy,flo) | 40.93 | 75.00 | 3.68 | 0.00 |
| mod3 | te(nh,tss) | 3.00 | 3.00 | 10.28 | 0.00 |
| mod4 | s(dec_time) | 7.08 | 8.10 | 3.04 | 0.00 |
| mod4 | s(flo) | 1.00 | 1.00 | 1.06 | 0.30 |
| mod4 | te(flo,dec_time) | 6.88 | 16.00 | 0.92 | 0.00 |
| mod4 | te(dec_time,doy,flo) | 41.24 | 75.00 | 3.82 | 0.00 |
| mod4 | s(nh) | 1.00 | 1.00 | 1.68 | 0.20 |
| mod4 | s(tss) | 1.00 | 1.00 | 20.71 | 0.00 |
| mod5 | s(dec_time) | 7.16 | 8.15 | 2.83 | 0.00 |
| mod5 | s(flo) | 1.00 | 1.00 | 0.81 | 0.37 |
| mod5 | te(flo,dec_time) | 6.37 | 16.00 | 0.82 | 0.00 |
| mod5 | te(dec_time,doy,flo) | 40.99 | 75.00 | 3.69 | 0.00 |
| mod5 | s(nh) | 2.94 | 3.68 | 0.61 | 0.68 |
| mod5 | te(nh,tss) | 2.29 | 20.00 | 0.85 | 0.00 |
| mod6 | s(dec_time) | 6.87 | 7.93 | 2.91 | 0.00 |
| mod6 | s(flo) | 1.00 | 1.00 | 0.55 | 0.46 |
| mod6 | te(flo,dec_time) | 6.81 | 16.00 | 0.90 | 0.00 |
| mod6 | te(dec_time,doy,flo) | 40.70 | 75.00 | 3.65 | 0.00 |
| mod6 | s(tss) | 3.80 | 4.79 | 6.06 | 0.00 |
| mod6 | te(nh,tss) | 0.17 | 20.00 | 0.01 | 0.26 |
| mod7 | s(dec_time) | 7.08 | 8.10 | 3.04 | 0.00 |
| mod7 | s(flo) | 1.00 | 1.00 | 1.06 | 0.30 |
| mod7 | te(flo,dec_time) | 6.88 | 16.00 | 0.92 | 0.00 |
| mod7 | te(dec_time,doy,flo) | 41.24 | 75.00 | 3.82 | 0.00 |
| mod7 | s(nh) | 1.00 | 1.00 | 1.68 | 0.20 |
| mod7 | s(tss) | 1.00 | 1.00 | 20.71 | 0.00 |
| mod7 | te(nh,tss) | 0.00 | 16.00 | 0.00 | 1.00 |
# summary stats of GAMs
map(mods3, ~ data.frame(
AIC = AIC(.x),
R2 = summary(.x)$r.sq)) %>%
enframe %>%
unnest %>%
mutate(smth_added = frmstab) %>%
select(name, smth_added, everything()) %>%
kable(digits = 2)| name | smth_added | AIC | R2 |
|---|---|---|---|
| mod1 | s(nh, bs = ‘tp’) | 868.68 | 0.59 |
| mod2 | s(tss, bs = ‘tp’) | 854.28 | 0.59 |
| mod3 | te(nh, tss, bs = c(‘tp’, ‘tp’)) | 843.63 | 0.60 |
| mod4 | s(nh, bs = ‘tp’) + s(tss, bs = ‘tp’) | 842.25 | 0.60 |
| mod5 | s(nh, bs = ‘tp’) + te(nh, tss, bs = c(‘tp’, ‘tp’)) | 845.49 | 0.60 |
| mod6 | s(tss, bs = ‘tp’) + te(nh, tss, bs = c(‘tp’, ‘tp’)) | 843.72 | 0.60 |
| mod7 | s(nh, bs = ‘tp’) + s(tss, bs = ‘tp’) + te(nh, tss, bs = c(‘tp’, ‘tp’)) | 842.25 | 0.60 |
Summary stats of best three three models:
# best model with season, year, flow
best3 <- map(mods3, ~ summary(.x)$r.sq) %>%
unlist %>%
which.max %>%
mods3[[.]]
best <- list(best1 = best1, best2 = best2, best3 = best3)
# smoother stats of GAMs
map(best, ~ summary(.x)$s.table %>% data.frame %>% rownames_to_column('smoother')) %>%
enframe %>%
unnest %>%
kable(digits = 2)| name | smoother | edf | Ref.df | F | p.value |
|---|---|---|---|---|---|
| best1 | s(dec_time) | 7.33 | 8.31 | 9.28 | 0.00 |
| best1 | s(doy) | 3.39 | 8.00 | 5.28 | 0.00 |
| best1 | te(dec_time,doy) | 6.64 | 15.00 | 1.42 | 0.00 |
| best2 | s(dec_time) | 6.45 | 7.55 | 4.65 | 0.00 |
| best2 | s(flo) | 3.20 | 4.05 | 1.93 | 0.09 |
| best2 | te(flo,dec_time) | 11.20 | 16.00 | 1.85 | 0.00 |
| best2 | te(dec_time,doy,flo) | 37.05 | 75.00 | 3.69 | 0.00 |
| best3 | s(dec_time) | 7.08 | 8.10 | 3.04 | 0.00 |
| best3 | s(flo) | 1.00 | 1.00 | 1.06 | 0.30 |
| best3 | te(flo,dec_time) | 6.88 | 16.00 | 0.92 | 0.00 |
| best3 | te(dec_time,doy,flo) | 41.24 | 75.00 | 3.82 | 0.00 |
| best3 | s(nh) | 1.00 | 1.00 | 1.68 | 0.20 |
| best3 | s(tss) | 1.00 | 1.00 | 20.71 | 0.00 |
| best3 | te(nh,tss) | 0.00 | 16.00 | 0.00 | 1.00 |
# summary stats of GAMs
map(best, ~ data.frame(
AIC = AIC(.x),
R2 = summary(.x)$r.sq)) %>%
enframe %>%
unnest %>%
kable(digits = 2)| name | AIC | R2 |
|---|---|---|
| best1 | 971.92 | 0.48 |
| best2 | 880.84 | 0.59 |
| best3 | 842.25 | 0.60 |
# predictions
sf_res3 <- map(best, function(x){
sf_mod %>%
mutate(
pred = predict(x)
)
}) %>%
enframe('mods') %>%
unnest
# plot
p <- ggplot(sf_res3, aes(x = date)) +
geom_point(data = sf_mod, aes(y = chl), size = 0.5) +
geom_line(aes(y = pred, colour = mods)) +
theme_bw() +
theme(
legend.position = 'top',
legend.title = element_blank()
)
ggplotly(p)